The spin degeneracy factor \(g_{L_{j}} = (2J_{L_{j}}+1) /[\prod \limits _{i=1}^3(2J_{h_i}+1)]\) .
The Dirac \(\delta \)  function guarantees the momentum conservation.
\(\mathcal {R}_{L_{j}}^{(x,p)}(\mathbf {x}_1,\mathbf {x}_2,\mathbf {x}_3;\mathbf {p}_1,\mathbf {p}_2,\mathbf {p}_3)\)  solving from the Wigner transformation [1]}, [2]} is
\(&& \mathcal {R}^{(x,p)}_{L_{j}}(\mathbf {x}_1,\mathbf {x}_2,\mathbf {x}_3;\mathbf {p}_1,\mathbf {p}_2,\mathbf {p}_3) = 8^2 e^{-\frac{(\mathbf {x}^{\prime }_1-\mathbf {x}^{\prime }_2)^2}{2\sigma _1^2}}e^{-\frac{2(\frac{m_1\mathbf {x}^{\prime }_1}{m_1+m_2}+\frac{m_2\mathbf {x}^{\prime }_2}{m_1+m_2}-\mathbf {x}^{\prime }_3)^2}{3\sigma _2^2}} \nonumber \\&&~~ \times e^{-\frac{2\sigma _1^2(m_2\mathbf {p}^{\prime }_{1}-m_1\mathbf {p}^{\prime }_{2})^2}{(m_1+m_2)^2\hbar ^2c^2}}e^{-\frac{3\sigma _2^2[m_3\mathbf {p}^{\prime }_{1}+m_3\mathbf {p}^{\prime }_{2}-(m_1+m_2)\mathbf {p}^{\prime }_{3}]^2}{2(m_1+m_2+m_3)^2\hbar ^2c^2}}. \) 
